Blending modes in Blender
Blend modes are implemented in the Color>"MixRGB" node, and are almost the same as in any layered 2D graphics software, like Photoshop or Gimp. 


  Premise
  
  Such Nodes don't act in the 3D
  world, but either on the 2D space of a surface's material or on the
  (still 2D) space of the rendered canvas. 
  
  In the 3D raytraced rendering, the color and intensity of each ray
  that hits the camera depends on its path through the scene's
  materials, that can be comprised of reflections, refraction,
  translucency, diffusion etc, in a way that is meant to be faithful to
  reality.
  
  In principle, you can't make two distinct objects "blend" their
  appearance in the 3D scene through 2D blending modes, i.e. by
  just arithmetically adding, subtracting, 
  averaging the colors of different layers.  If
  you're looking for a way to fake this behavior, then my answer isn't
  the right one for you.


To create a textured material for Cycles whose (2D) texture is obtained by blending different layers, you can follow these steps.


Select your object and assign an UV map to it. I've done this by selecting it (in my example, half a cylinder) in edit mode and hitting U. 


In the UV/Image editor window, open all the images that you will need as layers


Then create a new material and open the Node editor window.
Add new nodes by clicking "Add". You will need:


An "Input">"Texture Coordinate" node, and its "UV" socket to obtain information from the object's UV map
As many "Texture">"Image texture" nodes as your layers. Remember to connect their "Vector" input to the "UV" output from the Texture Coordinate node
Several "Color">"MixRGB" nodes to do the actual blending pairwise. They are displayed with different names in my screenshot, depending on the selected Blend mode.
Any "Shader" node (in this example, "Diffuse", that should be already present by default). This will take the Color output from the last MixRGB and its output will be plugged into the "Surface" socket of the Material.

Here's the complete node setup based on the example in your question 


And here's the rendered result